Elliptic finite-band potentials of a non-self-adjoint Dirac operator

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q≡Adn(x;m), where m∈(0,1) and A can be taken to be positive without loss of generality, for a non-self-adjoint Dirac operator L, which connects two well-known limiting cases of the plane wave (m=0) and of the sech potential (m=1). We show that, if A∈N, then the spectrum consists of R plus 2A Schwarz symmetric segments (bands) on iR. This characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space, and to appropriate connection problems for Heun's equation. Conversely, if A∉N, then the spectrum of L consists of infinitely many bands in C. When A∈N, the corresponding potentials generate finite-genus solutions for all the positive and negative flows associated with the focusing nonlinear Schrödinger hierarchy, including the modified Korteweg-deVries equation and the sine-Gordon equation.